The principle of general covariance can be fruitfully blended
with the concepts of local quantum field theory. This observation has,
during the past few years, led to considerable progress and new results in
quantum field theory in curved spacetime. I will review the central
concepts, and the results which have thus been achieved, such as
Spin-Statistics and PCT theorems and the completion of the perturbative
construction of $P(\phi)_4$ theories in quantum field theory on curved
spacetime. Moreover I will give some indications towards a possible
generalization of the concept of a generally covariant quantum field
theory which allows to consider quantum fields on Lorentzian
non-commutative spacetimes, with the prospect as a route to quantum
gravity.